I’m inclined to disagree. Deep abstraction gives us powerful tools for solving less abstract problems, including those that come out of the empirical sciences. Even fields developed with a deliberate eye to avoiding practical applications have sometimes turned out to make significant contributions to the sciences (I understand knot theory, for example, began this way, but has since turned out to have important applications in biochemistry).
You make a strong point, however; the question as to whether we can or cannot improve the efficiency of mathematical research appears to be an open one. I think that perhaps the real issue is that we don’t have a correct reductionist account of mathematics, and thus are not able to see clearly what we are doing when we build our theories. If we had a better road-map, I think that at the very least we could tie mathematics down to level 1/level 2 space so that we could have a better idea as to how we can measure the profitability of various possible lines of inquiry.
Well, my idea would be along the lines of thinking of mathematics as a combination of certain types of cognition combined with some sort of social feedback loop. We are phenomena and so are our actions. We do mathematics, therefore we should be able to make an empirical study of it.
I suppose that I would like an empirical dissection of mathematics as it is practiced by humans, something that would allow us to measure the statistical usefulness of various areas of mathematical thought. Do you think that this can’t be done? Do you think that it has already been done? If so I would be interested to hear your views, but I was under the impression that numerical cognition was still an open field. Or do you not think that this is related to numerical cognition?
I’m really not sure what to make of your reply. Even if you do
define mathematics as the study of systems with a complete reductionist account
shouldn’t we be able to give a reductionist account of the methods we use to give a reductionist account? This is one of those things that I do not feel I have a clear method for thinking around, yet it still seems like a problem, and a not-quite-mysterious one to boot. Intuitively it seems that we should be able to give a neurological account of mathematical thought and gain an empirical understanding of why some mathematics appears/is more applicable than others and what sorts of applications we might expect out of certain types of mathematical thought.′
Any insight you feel like sharing on this topic would be greatly appreciated. Am I just confused by some embedded mysterious question?
I’m inclined to disagree. Deep abstraction gives us powerful tools for solving less abstract problems, including those that come out of the empirical sciences. Even fields developed with a deliberate eye to avoiding practical applications have sometimes turned out to make significant contributions to the sciences (I understand knot theory, for example, began this way, but has since turned out to have important applications in biochemistry).
You make a strong point, however; the question as to whether we can or cannot improve the efficiency of mathematical research appears to be an open one. I think that perhaps the real issue is that we don’t have a correct reductionist account of mathematics, and thus are not able to see clearly what we are doing when we build our theories. If we had a better road-map, I think that at the very least we could tie mathematics down to level 1/level 2 space so that we could have a better idea as to how we can measure the profitability of various possible lines of inquiry.
What do you mean, we don’t have a correct reductionist account of mathematics?
I could define mathematics as the study of systems with a complete reductionist account.
Well, my idea would be along the lines of thinking of mathematics as a combination of certain types of cognition combined with some sort of social feedback loop. We are phenomena and so are our actions. We do mathematics, therefore we should be able to make an empirical study of it.
I suppose that I would like an empirical dissection of mathematics as it is practiced by humans, something that would allow us to measure the statistical usefulness of various areas of mathematical thought. Do you think that this can’t be done? Do you think that it has already been done? If so I would be interested to hear your views, but I was under the impression that numerical cognition was still an open field. Or do you not think that this is related to numerical cognition?
I’m really not sure what to make of your reply. Even if you do
shouldn’t we be able to give a reductionist account of the methods we use to give a reductionist account? This is one of those things that I do not feel I have a clear method for thinking around, yet it still seems like a problem, and a not-quite-mysterious one to boot. Intuitively it seems that we should be able to give a neurological account of mathematical thought and gain an empirical understanding of why some mathematics appears/is more applicable than others and what sorts of applications we might expect out of certain types of mathematical thought.′
Any insight you feel like sharing on this topic would be greatly appreciated. Am I just confused by some embedded mysterious question?